Linear Algebra, 2020a

Section IV. Matrix Operations 239
Above the arrows, the maps show that the two ways of going from V to X,
straight over via the composition or else in two steps by way of W, have the
same effect
⃗v g◦h
↦−→ g(h(⃗v)) ⃗v h
↦−→ h(⃗v)
g
↦−→ g(h(⃗v))
(this is just the definition of composition). Below the arrows, the matrices
indicate that multiplying GH into the column vector Rep B (⃗v) has the same
effect as multiplying the column vector first by H and then multiplying the
result by G.
Rep B,D (g ◦ h) =GH
Rep C,D (g) Rep B,C (h) =GH
As mentioned in Example 2.5, because the number of columns on the left
does not equal the number of rows on the right, the product as here of a 2×3
matrix with a 2×2 matrix is not defined.
(
)( )
−1 2 0 0 0
0 10 1.1 0 2
The definition requires that the sizes match because we want that the underlying
function composition is possible.
dimension n space
h
−→ dimension r space
g
−→ dimension m space
(∗)
Thus, matrix product combines the m×r matrix G with the r×n matrix F to
yield the m×n result GF. Briefly: m×r times r×n equals m×n.
2.8 Remark The order of the dimensions can be confusing. In ‘m×r times r×
n equals m×n’ the number written first is m. But m appears last in the map
dimension description line (∗) above, and the other dimensions also appear in
reverse. The explanation is that while h is done first, followed by g, we write
the composition as g ◦ h, with g on the left (arising from the notation g(h(⃗v))).
That carries over to matrices, so that g ◦ h is represented by GH.
We can get insight into matrix-matrix product operation by studying how
the entries combine. For instance, an alternative way to understand why we
require above that the sizes match is that the row of the left-hand matrix must
have the same number of entries as the column of the right-hand matrix, or else
some entry will be left without a matching entry from the other matrix.
Another aspect of the combinatorics of matrix multiplication, in the sum
defining the i, j entry, is brought out here by the boxing the equal subscripts.
p i,j = g i, 1 h 1 ,j + g i, 2 h 2 ,j + ···+ g i, r h r ,j
The highlighted subscripts on the g’s are column indices while those on the h’s
are for rows. That is, the summation takes place over the columns of G but